Let $D$ be a family of $k$-subsets (called blocks) of a $v$-set
$X(v)$. Then $D$ is a $(v,k,t)$ covering design or covering if every
$t$-subset of $X(v)$ is contained in at least one block of $D$. The
number of blocks is the size of the covering, and the minimum size of
the covering is called the covering number. In this paper we consider
the case $t=2$, and find several infinite classes of covering numbers.
We also give upper bounds on other classes of covering numbers.